A105641 Number of hill-free Dyck paths of semilength n, having no UUDD's, where U=(1,1) and D=(1,-1) (a hill in a Dyck path is a peak at level 1).

0, 1, 2, 5, 14, 39, 111, 322, 947, 2818, 8470, 25677, 78420, 241061, 745265, 2315794, 7228702, 22656505, 71273364, 224965675, 712249471, 2261326010, 7197988973, 22966210236, 73437955105, 235307698544, 755395560220, 2429293941019

Offset: 2

Author

Emeric Deutsch, May 08 2006

Keywords

• nonn

Comments

a(n) = A105640(n,0).

Examples

			a(4)=2 because we have UUDUDUDD and UUUDUDDD.
		

Links

• E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

Crossrefs

Cf. A118995.

Programs

Maple

G:=((1+z)^2-sqrt((1+z^2)^2-4*z))/2/z/(2+z+z^2)-1: Gser:=series(G,z=0,36): seq(coeff(Gser,z^n),n=2..32);

Formula

G.f.: [(1+z)^2-sqrt((1+z^2)^2-4z)]/[2z(2+z+z^2)]-1.

D-finite with recurrence 2*(n+1)*a(n) +(-7*n+5)*a(n-1) +(n-5)*a(n-2) +2*(-n-1)*a(n-3) +2*(2*n-7)*a(n-4) +(n-5)*a(n-5) +(n-5)*a(n-6)=0. - R. J. Mathar, Jul 24 2022